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AAEC 4302 ADVANCED STATISTICAL METHODS IN AGRICULTURAL RESEARCH

AAEC 4302 ADVANCED STATISTICAL METHODS IN AGRICULTURAL RESEARCH. Chapter 11: Sampling Theory in Regression Analysis. Statistical Inference.

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AAEC 4302 ADVANCED STATISTICAL METHODS IN AGRICULTURAL RESEARCH

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  1. AAEC 4302ADVANCED STATISTICAL METHODS IN AGRICULTURAL RESEARCH Chapter 11: Sampling Theory in Regression Analysis

  2. Statistical Inference • Recall that the parameter estimates obtained by applying the OLS formulas are not equal to the true (population) model parameters . • How close the estimated value of a given parameter is, to its true, unknown population value?

  3. The Normal Regression Model • uiis a random variable with normal probability distribution E(ui) = 0 and σ(ui) = σu • Yi σ(Yi) = σu

  4. The Normal Regression Model • ui is normally distributed with E(ui)=0 and σ(ui)= 5 • If Xi = 5, What can you say about Yi?

  5. The Normal Regression Model • Two assumptions about the relations among the disturbances for different observations: • The ui are independent • The value of the disturbance from one observation in no way affects the value that occurs for another. • σ(ui) = σuthe same for all observations

  6. The Normal Regression Model Yi E[Y2]=91 Y2=89 E[Yi]=β0+β1X1 Y1=69 E[Y1]=67 X X=5 X=7 Sample

  7. The Normal Regression Model

  8. Sampling Distribution of OLS Formulas • Monte Carlo experiments • Those estimated parameter values represent the probability distribution of the OLS estimator for

  9. Sampling Distribution of OLS Formulas • In the simple linear regression model: 2 æ ö æ ö ΣX ç s ÷ ç ÷ 2 β ~ N β , i ç ÷ ç ÷ ( ) 2 0 0 - n X X å è ø è ø i

  10. Calculating the S.E. of the Estimators • The standard error of the estimator is the standard error of . • The expression , which appears in is known as the total variation in X.

  11. Calculating the S.E. of the Estimators Example: • σu =5, β0 =7 and β1 =12 • Assume the total variation in X equals 9

  12. Calculating the S.E. of the Estimators • For a set of data for which total variation in X is equal to 25 • Standard Error for this case σ(β1) = 1 • Pr (11≤β1≤13) = ? When Standard Error is smaller there is a greater possibility that est. β1 will take on a value in some interval centered around true β1 valueThe smaller the standard error, the more precise is est. β1 as an estimator of β1

  13. Interval Estimation • EARNSi = β0 + β1 EDi + ui • Estimated model EARNSi = -1.315+ 0.79 EDi (1.540) (0.128) R2=0.285 SER=4.361

  14. Interval Estimation α/2 = Pr ( ≥ + h) = Pr (Z ≥ ), where = h/σ( ) = Pr(Z ≥ h/σ( )) Pr (Z ≥ Zc) = α/2 h = Zcσ( ) ± h, where h = tc s( ), where tc is determined from Pr (t ≥ tc) = α/2

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